Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?
The volume of a cone (V) with radius (r) and height (h) is given by,
\begin{align}V=\frac{1}{3}\pi r^2h\end{align}
It is given that,
\begin{align}h=\frac{1}{6} r\Rightarrow r =6h\end{align}
\begin{align}\therefore V=\frac{1}{3}\pi (6h)^2.h = 12\pi h^3\end{align}
The rate of change of volume with respect to time (t) is given by,
\begin{align} \frac{dV}{dt}=12 \pi \frac{d}{dh}(h^3).\frac{dh}{dt}[By\; Chain\; Rule]\end{align}
\begin{align}=12 \pi (3h^2).\frac{dh}{dt}\end{align}
\begin{align}=36 \pi h^2.\frac{dh}{dt}\end{align}
It is also given that
\begin{align}\frac{dV}{dt}=12\;cm^3/s \end{align}
Therefore, when h = 4 cm, we have:
\begin{align}12=36\pi (4)^2.\frac{dh}{dt}\end{align}
\begin{align}\Rightarrow \frac{dh}{dt}=\frac{12}{36\pi (16)}=\frac{1}{48\pi}\end{align}
Hence, when the height of the sand cone is 4 cm, its height is increasing at the rate of
\begin{align}\frac{1}{48\pi}.\end{align}
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i) f : R → R defined by f(x) = 3 – 4x
(ii) f : R → R defined by f(x) = 1 + x2
Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.
Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
The rate of change of the area of a circle with respect to its radius r at r = 6 cm is
(A) 10π (B) 12π (C) 8π (D) 11π
Answer the following as true or false.
\begin{align}(i) \overrightarrow{a}\; and\; \overrightarrow{-a}\; are\; collinear.\end{align}
(ii) Two collinear vectors are always equal in magnitude.
(iii) Two vectors having same magnitude are collinear.
(iv) Two collinear vectors having the same magnitude are equal.
Let f : R → R be defined as f(x) = 3x. Choose the correct answer.
(A) f is one-one onto
(B) f is many-one onto
(C) f is one-one but not onto
(D) f is neither one-one nor onto.